What is the Integral of (Tan x)^2? Learn the Solution Here!

[transgalactic]

what is the integral of (tan x)^2

i added a file in which i showed the way i tried to solve it

 

[Mathgician]

what is its identity?

 

[transgalactic]

the integral is:

(tan x)^2

 

[Mathgician]

I'm going to ask again, what is its direct trig identity

 

[Mathgician]

(tanx)^2=[(secx)^2]-1

 

[transgalactic]

the trig identity is tangent

but my integral is
tangent square

 

[Gib Z]

Lol

Are you familiar with what Mathgician said? Its a well known Trigonometric Identity, $\tan^2 x = \sec^2 x -1$

Knowing that,

$$\int \tan^2 x dx = \int (\sec^2 x - 1) dx = \int \sec^2 x dx - \int 1 dx = (\int \sec^2 x dx ) - x$$

For the integral of (sec x) squared, if you don't already know it, try differentiating tan x, what is that?

 

[Mathgician]

If you are doing integrals right now, you have to have come across the derivative of tanx from differential calc class. No offense, but its the most basic derivative and identity is so common, I just thought you already should have known. I've seen you help many people on this board with integrals, I assumed that you would already knew or maybe you just had a brain fart or something. Its alright I have moments like these also.

 

[Gib Z]

In fact, I just look through your attachment right there. You didn't even need Mathgicians or my suggestions for that identity, you got it yourself! 1/cos^2 x is sec^2 x :D You just needed to notice that was the derivative of tan x, instead of using cos^2 x = (1+cos 2x)/2

 

[transgalactic]

yeeepppp thank you very much

i got it
it was right under my nose